What is wrong with the following argument?
By Postulate 3, we describe a circle centered at $A$ with a distance $BC$. Pick any point on the boundary of the circle. Call this point $F$. By Postulate 1, we connect $A$ and $F$ in a straight line. By the def. of circles, $AF$ = $BC$.
Best Answer
Postulate 3 allows you to construct a circle centered on one end of the given length (to be used as the radius). It does not justify constructing a circle centered on a point that is not an endpoint of the given segment.
Consequently, in your first line, one of $B$ or $C$ must be $A$.